Preheat oven to 350 F.

Spread bread cubes out on rimmed baking sheet. Toast until beginning to feel dry but not hard, about 7 minutes. Cool on sheet. Whisk eggs, 3/4 cup sugar, and 1/8 teaspoon salt in large bowl to blend. Bring cream and milk to simmer in medium saucepan. Very gradually whisk cream mixture into egg mixture. Strain custard into medium bowl; mix in vanilla. Cool to lukewarm, about 30 minutes.

Whisk brown sugar and butter in large skillet over medium-high heat until smooth sauce forms, about 1 minute. Whisk in rum, lemon juice, and 1/4 teaspoon salt. Add bananas and toss just until coated, about 30 seconds. Remove from heat.

Lightly butter 8×8x2-inch glass baking dish (8-cup capacity). Scatter 1/3 of bread in dish. Top with half of bananas and half of syrup from skillet. Scatter half of remaining bread over. Top with remaining bananas and syrup from skillet, then remaining bread. Pour custard over. Let pudding stand 30 minutes, occasionally pressing down bread to submerge and to absorb custard.

Meanwhile, get the oven back to 350 F. Place dish with pudding in 13×9x2-inch metal baking pan. Add enough warm water to pan to reach halfway up sides of pudding dish. Cover baking pan with foil. Bake pudding 30 minutes. Uncover and bake until puffed and set in center, about 45 minutes longer. Remove from water bath and let stand 15 minutes. Serve warm or at room temperature with warm caramel sauce.